Approximate maximin share allocation for indivisible goods under a knapsack constraint

The maximin share (MMS) allocation problem under a knapsack constraint is to allocate a set of indivisible goods to a set of n heterogeneous agents, such that the total cost of the allocated goods does not exceed the given budget, and the approximation ratio of the MMS allocation is as large as possible. For any $$\epsilon \in (0, 1)$$ ϵ ∈ ( 0 , 1 ) , we prove that $$(\frac{93}{95}+ \epsilon )$$ ( 93 95 + ϵ ) -approximate MMS allocation does not always exist for two agents, while the MMS allocation problem without a knapsack constraint always has an MMS allocation for two agents. We propose a bag-filling based algorithm that can produce a $$\frac{n}{3n-2}$$ n 3 n - 2 -approximate MMS allocation. When $$n=2$$ n = 2 and $$n=3$$ n = 3 , by more careful analysis, we improve the approximation ratios to $$\frac{2}{3}$$ 2 3 and $$\frac{1}{2}$$ 1 2 , respectively.